Theorem lspsnsub

Description: Swapping subtraction order does not change the span of a singleton. (Contributed by NM, 4-Apr-2015)

	Ref	Expression
Hypotheses	lspsnsub.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
	lspsnsub.s	⊢ − = ( -g ‘ 𝑊 )
	lspsnsub.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
	lspsnsub.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
	lspsnsub.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
	lspsnsub.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
Assertion	lspsnsub	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) = ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) )

Proof

Step	Hyp	Ref	Expression
1		lspsnsub.v	⊢ 𝑉 = ( Base ‘ 𝑊 )
2		lspsnsub.s	⊢ − = ( -g ‘ 𝑊 )
3		lspsnsub.n	⊢ 𝑁 = ( LSpan ‘ 𝑊 )
4		lspsnsub.w	⊢ ( 𝜑 → 𝑊 ∈ LMod )
5		lspsnsub.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑉 )
6		lspsnsub.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑉 )
7	1 2	lmodvsubcl	⊢ ( ( 𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
8	4 5 6 7	syl3anc	⊢ ( 𝜑 → ( 𝑋 − 𝑌 ) ∈ 𝑉 )
9		eqid	⊢ ( invg ‘ 𝑊 ) = ( invg ‘ 𝑊 )
10	1 9 3	lspsnneg	⊢ ( ( 𝑊 ∈ LMod ∧ ( 𝑋 − 𝑌 ) ∈ 𝑉 ) → ( 𝑁 ‘ { ( ( invg ‘ 𝑊 ) ‘ ( 𝑋 − 𝑌 ) ) } ) = ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) )
11	4 8 10	syl2anc	⊢ ( 𝜑 → ( 𝑁 ‘ { ( ( invg ‘ 𝑊 ) ‘ ( 𝑋 − 𝑌 ) ) } ) = ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) )
12		lmodgrp	⊢ ( 𝑊 ∈ LMod → 𝑊 ∈ Grp )
13	4 12	syl	⊢ ( 𝜑 → 𝑊 ∈ Grp )
14	1 2 9	grpinvsub	⊢ ( ( 𝑊 ∈ Grp ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ) → ( ( invg ‘ 𝑊 ) ‘ ( 𝑋 − 𝑌 ) ) = ( 𝑌 − 𝑋 ) )
15	13 5 6 14	syl3anc	⊢ ( 𝜑 → ( ( invg ‘ 𝑊 ) ‘ ( 𝑋 − 𝑌 ) ) = ( 𝑌 − 𝑋 ) )
16	15	sneqd	⊢ ( 𝜑 → { ( ( invg ‘ 𝑊 ) ‘ ( 𝑋 − 𝑌 ) ) } = { ( 𝑌 − 𝑋 ) } )
17	16	fveq2d	⊢ ( 𝜑 → ( 𝑁 ‘ { ( ( invg ‘ 𝑊 ) ‘ ( 𝑋 − 𝑌 ) ) } ) = ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) )
18	11 17	eqtr3d	⊢ ( 𝜑 → ( 𝑁 ‘ { ( 𝑋 − 𝑌 ) } ) = ( 𝑁 ‘ { ( 𝑌 − 𝑋 ) } ) )